For Peer Review Only - Cornell UniversityFor Peer Review Only September 26, 2015 Combustion Theory...

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LES/PDF for Premixed Combustion in the DNS Limit

Journal: Combustion Theory and Modelling

Manuscript ID TCTM-2015-09-103

Manuscript Type: Original Manuscript

Date Submitted by the Author: 26-Sep-2015

Complete List of Authors: Pope, Stephen; Cornell University, Sibley School of Mechanical and Aerospace Engineering Tirunagari, Ranjith; Cornell University, Sibley School of Mechanical & Aerospace Engineering

Keywords: PDF Methods, large-eddy simulation, turbulent combustion, DNS limit, premixed flames

URL: http://mc.manuscriptcentral.com/tctm E-mail: [email protected]

Combustion Theory and Modelling

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September 26, 2015 Combustion Theory and Modelling DNSLimit˙v3

Combustion Theory and ModellingVol. 00, No. 00, Month 200x, 1–28

RESEARCH ARTICLE

LES/PDF for Premixed Combustion in the DNS Limit

Ranjith R. Tirunagaria and Stephen B. Popea∗

aSibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY

14853, USA

(received Date Month 2015)

We investigated the behavior of the composition probability density function (PDF) modelequations used in a large eddy simulation (LES) of turbulent combustion in the DNS limit;that is, in the limit of the LES resolution length scale ∆ (and the numerical mesh spacingh) being small compared to the smallest flow length scale, so that the resolution is sufficientto perform a direct numerical simulation (DNS). The correct behavior of a PDF model inthe DNS limit is that the resolved composition fields satisfy the DNS equations, and thereare no residual fluctuations (i.e., the PDF is everywhere a delta function). In the DNS limit,the treatment of molecular diffusion in the PDF equations is crucial, and both the “random-walk” and “mean-drift” models for molecular diffusion are investigated. Two test cases areconsidered, both of premixed laminar flames (of thickness δL). We examine the solutions ofthe model PDF equations for these test cases as functions of ∆/δL and h/δL. Each of thetwo PDF models has advantages and disadvantages. The mean-drift model is consistent withthe DNS limit, but it is more difficult to implement and computationally more expensive.The random-walk model is not consistent with the DNS limit in that it produces non-zeroresidual fluctuations. However, if the specified mixing rate Ω normalized by the reaction timescale τc is sufficiently large (Ωτc & 1), then the residual fluctuations are less than 10% andthe observed flame speed and thickness are close to their laminar values. Away from the DNSlimit (i.e., h/δL & 1), the observed flame thickness scales with the mesh spacing h, and theflame speed scales with Ωh. For this case it is possible to construct a non-general specificationof the mixing rate Ω such that the flame speed matches the laminar flame speed.

Keywords: PDF methods, large-eddy simulation, turbulent combustion, DNS limit,premixed flames

∗Corresponding author. Email: [email protected]

ISSN: 1364-7830 print/ISSN 1741-3559 onlinec© 200x Taylor & FrancisDOI: 10.1080/1364783YYxxxxxxxxhttp://www.tandfonline.com

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2 R.R. Tirunagari and S.B. Pope

Nomenclature

Roman

CD mixing rate constant, Eq. 28CL mixing rate constant, Eq. 36Cm mixing rate constant, Eq. 9D nozzle diameterD molecular diffusivityDe effective diffusivity, Eq. 6Dn numerical diffusivity, Eq. 30Dr residual diffusivity

f(ψ;x, t) density-weighted PDF of compositionh mesh spacingN number of particlesS reaction source termsL laminar flame speedT temperaturet timeU fluid velocityuF flame speedv specific volumeW(t) isotropic Wiener processX∗(t) particle positionx positionYOH mass fraction of OHz coordinate along the axis of the opposed nozzles

Greek

∆ LES resolution length scaleδ smallest flow length scaleδF flame thickness, Eq. 26δL laminar flame thicknessρ fluid densityσφ,max maximum residual standard deviationτc reaction time scaleφ(x, t) compositionψ sample-space variables corresponding to φΩ mixing rate, Eq. 9

Superscripts

φ∗ particle composition

φ resolved composition

φ∗

resolved composition at particle position

Subscripts

Tb burnt temperatureTu unburnt temperature

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Combustion Theory and Modelling 3

1. Introduction

Probability density function (PDF) methods[1] are seeing increased use in con-junction with large eddy simulations (LES) as an effective means of accountingfor the turbulence-chemistry interactions occurring in turbulent combustion[2, 3].While most previous applications of LES/PDF have been to non-premixedcombustion[4–6], recently there have been several applications to premixed tur-bulent combustion[7–9].In large-eddy simulations[10–12], modeled equations are solved for the larger-

scale resolved motions, while the effects of the smaller-scale (unresolved) residual

motions are modeled. A specified LES resolution length scale, ∆, demarcates theresolved and residual scales. The most common approach to LES is to define theresolved fields by a filtering operation, with ∆ being the characteristic width ofthe filter[10, 13]. However, as discussed below, for the present considerations, it ispreferable to define the resolved fields in terms of conditional means[12, 14].In practice, the resolution length scale ∆ is usually linked to the mesh spacing

h used in the numerical solution of the LES equations, e.g. ∆ = h or ∆ = 2h,which leads to a mingling of modeling and numerical errors[10, 15]. However, it isimportant to retain the distinction between these two quantities, in particular sothat numerically-accurate solutions can be considered for h/∆ ≪ 1.An obviously-desirable property of an LES formulation is that it correctly con-

verges to the DNS limit. That is, as the resolution scale ∆ becomes small comparedto the smallest flow length scale δ (so that the residual motions tend to zero), thenthe LES equations should tend to the fundamental conservation laws (e.g., theNavier-Stokes equations) that apply in a direct numerical simulation (DNS). Thepurpose of this paper is to examine the behavior of the composition PDF modelsin this DNS limit.In a typical application of LES to a high-Reynolds-number flow, ∆/δ is large, and

molecular transport has a negligible direct effect on the resolved fields: its primaryeffects are the dissipation of kinetic energy and the mixing of species below theresolved scale. For this reason, in LES, molecular transport is often neglected ortreated in an unrealistically simplified way[16]. But in the DNS limit, moleculartransport is a dominant process, and hence must be implemented correctly if theDNS limit is to be attained.We now provide three reasons why it is important for LES/PDF models to con-

verge correctly to the DNS limit.First, in laboratory flames, the molecular diffusivity can be much larger than one

might expect, given the typical flow Reynolds numbers of order 10,000. This is pri-marily because the molecular diffusivity D (of dimensions length squared dividedby time) increases strongly with temperature T (e.g., as T 1.7), and hence at flametemperatures can be 30 times its value at room temperature. Kemenov & Pope[16]show that, in a typical LES of the Barlow & Frank flames[17], the molecular dif-fusivity is generally several times the residual diffusivity for temperatures above1,000K.Second, there is recent interest in premixed combustion in turbulent counter-

flow burners[9, 18]. When LES is applied to these flames, it is typically found thatthe resolution scale ∆ is of the same order of magnitude as the laminar flamethickness, δL. Hence, the direct effects of molecular transport are important, as isthe convergence to the DNS limit.Third, as computer power continues to increase, it is becoming practicable to per-

form high-fidelity LES for a larger class of flows. Whereas in conventional LES, theresolution scale ∆ is chosen to resolve only the larger, energy-containing motions,

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in high-fidelity LES, ∆ is chosen to be much smaller to provide partial resolutionof the smallest-scale processes[19]. For many occurrences of turbulent combustion,in which the rate-controlling processes are at the smallest scales, high-fidelity LESmay be required in order to provide reliable simulations[3]. By definition, high-fidelity LES approaches the DNS limit, and hence for this purpose it is essentialthat the LES/PDF models used converge correctly in this limit.The effects of molecular diffusion on the joint PDF of compositions can be de-

composed into: spatial transport which affects the mean compositions; and, mixing,which decreases the composition variances, while not affecting the means. Thesetwo effects are modeled separately. Here it is sufficient to consider the simplest mix-ing model, namely the “interaction by exchange with the mean” (IEM) model[20],which (like other models) causes the mixing to occur at a specified rate Ω. Molecu-lar transport can be implemented in two ways, and we consider both, with detailsgiven in Sec. 2. These are referred to as the Random-Walk (RW) model and theMean-Drift (MD) model.In the DNS limit, the residual diffusivity Dr vanishes in comparison to the molec-

ular diffusivity D. However, the mixing rate Ω does not vanish, but remains as animportant parameter.We investigate the DNS limit by applying the LES/PDF model equations to two

test cases, both of premixed laminar flames. The first is an idealized, unstrained,one-dimensional, freely-propagating, premixed laminar flame, with one-step chem-istry. For this case, the LES/PDF equations are solved using a simple 1D MATLABcode. The second case is a strained methane/air premixed laminar flame in an op-posed jet configuration, in which a cold methane/air jet is opposed to a jet ofhot combustion products. In this case the LES/PDF equations are solved usingthe same 3D code used in many previous applications of LES/PDF to turbulentflames, and the chemistry is described by a 16-species reduced mechanism.The LES/PDF equations are solved by particle-mesh methods, using N particles,

mesh spacing h, and time steps ∆t. Except where mentioned explicitly otherwise,the number of particles can be considered to be sufficiently large, and the time stepsufficiently small, so that the only significant numerical errors arise from the meshspacing, h.The focus of the investigation is on the solution of the LES/PDF equations as a

function of ∆/δL, i.e., the ratio of the resolution length scale to the laminar flamethickness: the DNS limit corresponds to this ratio tending to zero. We considerboth the numerically-accurate solutions (corresponding to small h/∆), and alsonumerical solutions with h/∆ of order unity.The remainder of the paper is organized as follows. In Sec. 2 we describe the two

LES/PDF models considered, namely the random-walk model and the mean-driftmodel. In Secs. 3 we describe the first of the two test problems considered, namelyan idealized, unstrained, one-dimensional, freely-propagating laminar, premixedflame. Numerically-accurate solutions to the LES/PDF equations in the DNS limitare presented and discussed. These solutions are obtained using a mesh-free parti-cle method that is described in Appendix A. Then, we present numerical solutionsobtained using a conventional cloud-in-cell (CIC) particle/mesh method, which isdescribed in Appendix B. In this case, the solutions depend on the specified nor-malized mesh size h/∆. In Sec. 4 we describe the second test case of a strainedmethane/air opposed-jet, laminar, premixed flame. The numerically-accurate so-lution of the laminar-flow equations for this case is obtained using CHEMKIN-PRO[21]. The numerical solutions to the LES/PDF equations in the DNS limitare obtained using the same 3D particle/mesh method used in many previousLES/PDF studies (e.g., [6, 9]). Based on the results obtained and other consider-

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ations, in Sec. 5 we discuss the relative merits of the random-walk and mean-driftmodels. Conclusions are drawn in Sec. 6.

2. Composition PDF Methods

2.1 Governing equations

We consider the standard case of a low-Mach-number, single-phase, turbulent re-acting flow. As functions of position, x, and time, t, the flow is described by thefields of velocity, U(x, t), and composition, φ(x, t). In general, the nφ componentsof the composition vector φ can be taken to be the mass fractions of the ns chemicalspecies and the enthalpy. For the low-Mach-number flow considered, as far as thethermochemistry is concerned, the pressure can be approximated as being constantand uniform, and is not shown explicitly in the notation. Thus, the equations ofstate determining the density ρ and temperature T are of the form

ρ(x, t) = ρ(φ(x, t)), (1)

and

T (x, t) = T (φ(x, t)). (2)

The velocity field is governed by the usual equations for the conservation of massand momentum. The focus here is on the composition field, which we take to begoverned by the equation

ρDφ

Dt= ρ

∂φ

∂t+ ρU · ∇φ = ∇ · (ρD∇φ) + ρS, (3)

where S is the reaction source term (i.e., the rate of change of φ due to chemicalreactions), and D is the molecular diffusivity. Both S and D are known functions ofφ. The form of the diffusion term embodies the assumption of unity Lewis numbers,which is discussed further below.

2.2 LES/PDF formulation

In an LES/PDF simulation of the flow considered, the primary quantities involved

are the resolved density, 〈ρ(x, t)〉, the density-weighted resolved velocity, U(x, t),

and the density-weighted PDF, f(ψ;x, t), where ψ are the sample-space variablescorresponding to φ. This PDF may be viewed as the PDF of composition, condi-tional on the resolved velocity field[12, 14].While a decomposition of the composition field into resolved and residual com-

ponents is not needed, it is nevertheless useful to consider the (density-weighted)resolved composition field defined as

φ(x, t) ≡

∫f(ψ;x, t)ψ dψ, (4)

where, here and below, the integration is over the whole of the composition space.The PDF then represents the residual fluctuations about this mean.For some purposes, it is more convenient to use the specific volume v ≡ 1/ρ

rather than the density. For example, the resolved density is obtained from the

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PDF by

〈ρ(x, t)〉−1 = v(x, t) =

∫f(ψ;x, t)v(ψ) dψ, (5)

where v ≡ 1/ρ is the equation of state for specific volume.

2.3 Lagrangian particle method

In PDF methods, the modeling and numerical solution are usually performed us-ing a Lagrangian particle method[1, 2, 10]. The position of the general particleis denoted by X∗(t), and its composition by φ∗(t). The modeling is performedby specifying the evolution of these particle properties; and the PDF considered,f(ψ;x, t), is the density-weighted PDF of φ∗(t) conditional on X∗(t) = x.

2.4 Random-walk model

In the random-walk (RW) model, the particle is specified to move with the resolvedvelocity, plus a random walk, the magnitude of which is determined by the mean

effective diffusivity (which has dimensions of length squared divided by time). Themolecular diffusivity is D, and part of the LES modeling is to define a residual dif-

fusivity, Dr, which accounts for transport due to the residual motions. The effectivediffusivity is then defined as their sum:

De ≡ D +Dr. (6)

According to the RW model, the particle position evolves by the stochastic dif-ferential equation (SDE)

dX∗ = [U+ v∇(〈ρ〉De)]∗ dt+ (2D

∗

e)1/2 dW, (7)

where W(t) is an isotropic Wiener process. Note that D∗

e denotes the effective

diffusivity based on the particle composition (and Dr); whereas D∗

e denotes themean effective diffusivity evaluated at the particle location, i.e., it is a shorthandnotation for De(X

∗(t), t).The evolution equation for the particle composition is:

dφ∗

dt= −Ω∗(φ∗ − φ

∗

) + S(φ∗). (8)

On the right-hand side, the first term is the IEM mixing model, and the secondterm is the reaction source term. The standard model for the mixing rate, Ω, is

Ω = CmDe

∆2, (9)

where Cm is a model constant.(Note that, in Eqs. 7 and 9, the diffusion coefficient and the mixing rate are

based on the mean effective diffusivity, D∗

e. One can consider three additional modelvariants in which one or both of these quantities is instead based on the particleeffective diffusivity, D∗

e . Tests (for the case described in Sec. 3) show that there are

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no qualitative differences between the four variants, and only modest quantitativedifferences.)The PDF equation deduced from these particle equations is:

〈ρ〉∂f

∂t+ 〈ρ〉U ·∇f = ∇· (〈ρ〉De∇f)−

∂

∂ψα

(〈ρ〉f

[−Ω(ψα − φα) + Sα(ψ)

]), (10)

where the summation convention applies to the repeated composition suffix α.Multiplying this PDF equation by ψ and integrating over the composition space,

we obtain the implied conservation equation for the resolved composition:

〈ρ〉∂φ

∂t+ 〈ρ〉U · ∇φ = ∇ · (〈ρ〉De∇φ) + 〈ρ〉S. (11)

Note that the random walk (and in particular the drift term in Eq. 7) is constructedto yield the diffusion term in Eq. 11; and, by construction, the IEM model doesnot (directly) affect the resolved composition φ, and hence the mixing rate Ω doesnot appear in Eq. 11.

2.5 Mean-drift model

In the random walk model, molecular transport is implemented via the randomwalk in the SDE for the particle position,X∗(t). In contrast, in the mean drift (MD)model[22, 23], molecular transport is implemented in the ODE for composition. Theparticle evolution equations for the MD model are:

dX∗ = [U+ v∇(〈ρ〉Dr)]∗ dt+ (2D

∗

r)1/2 dW, (12)

dφ∗

dt= [v∇ · (〈ρ〉D∇φ)]∗ − Ω∗(φ∗ − φ

∗

) + S(φ∗). (13)

Thus, in the SDE for position (Eq. 12) the random walk is based on Dr, andaccounts solely for the transport due to the residual motions; and in the particlecomposition equation (Eq. 13), the first term of the right-hand side accounts fortransport by molecular diffusion.For the mean-drift model, the corresponding PDF equation (deduced from

Eqs. 12–13) is:

〈ρ〉∂f

∂t+〈ρ〉U·∇f = ∇·(〈ρ〉Dr∇f)−

∂

∂ψα

(〈ρ〉f

[v∇ · (〈ρ〉D∇φα)− Ω(ψα − φα) + Sα(ψ)

]).

(14)The corresponding equation for the resolved composition is again Eq. 11, i.e., it isidentical to that for the random-walk model.

2.6 Residual covariance and the DNS limit

The residual covariance is defined as

φ′′αφ′′

β ≡

∫f(ψ) (ψα − φα)(ψα − φβ) dψ. (15)

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With the viewpoint taken here, that f is the PDF of the composition conditionalon the resolved velocity field, in the DNS limit, this PDF is a delta function, i.e.,

f(ψ;x, t) = δ(ψ − φ(x, t)). (16)

Consequently, the residual covariance is zero.Note that, if the alternative, filtering viewpoint were taken, then in the DNS

limit, the residual covariance would instead be non-zero. With the usual definitionof the filter width, to leading order in ∆, the residual covariance is

φ′′αφ′′

β =∆2

12∇φα · ∇φβ (17)

(see Eq.13.157 of [10]). For the present purposes, it is clearly preferable to adoptthe conditional PDF viewpoint so that the residual covariance is zero in the DNSlimit.For the mean-drift model, the evolution equation for the residual covariance

(deduced from Eq. 14) is:

〈ρ〉∂

∂tφ′′αφ

′′

β + 〈ρ〉U · ∇φ′′αφ′′

β = ∇ · (〈ρ〉Dr∇φ′′αφ′′

β) (18)

+ 2〈ρ〉Dr∇φα · ∇φβ − 2〈ρ〉Ω φ′′αφ′′

β

+ 〈ρ〉φ′′αSβ + 〈ρ〉φ′′βSα.

In the DNS limit, the residual covariance and Dr are zero, and hence the right-hand side of this equation is zero. Thus there is no mechanism causing the residualcovariance to depart from zero.For the random-walk model, the residual covariance equation is almost the same,

but with the effective diffusivity De in place of the residual diffusivity Dr in thefirst two terms on the right-hand side. But this difference is crucial, because, inthe DNS limit, the second term on the right-hand side is then non-zero, and yieldsthe spurious production term 2〈ρ〉D∇φα · ∇φβ . This spurious production causesthe residual variances to depart from zero (at locations where there are non-zerocomposition gradients).It is interesting to observe that, with the random-walk model, if the residual

covariance is determined by a balance between the spurious production and thedissipation term (involving Ω), then this yields

φ′′αφ′′

β =D

Ω∇φα · ∇φβ . (19)

This is consistent with Eq. 17 if Ω is determined by Eq. 9, with Cm = 12. (We donot ascribe any significance to this observation.)

3. Idealized, Unstrained, Plane, Premixed Laminar Flame

3.1 Definition of the test case

The first test case we consider is that of an idealized, unstrained, one-dimensional,plane, premixed laminar flame. The single direction of variation is denoted by x.

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The chemistry is described by a single composition variable, φ(x, t), which maybe considered to be a normalized product mass fraction, or a reaction progressvariable. At x = −∞ there are pure reactants (φ = 0), and at x = ∞ there arecompletely burnt products (φ = 1). In a frame fixed with the unburnt reactants,the flame propagates to the left at the flame speed uF . Alternatively, in the framefixed to the flame, the reactants flow to the right at speed uF .The very simple thermochemistry is defined in terms of φ as follows. The tem-

perature T is specified as

T = Tu + φ(Tb − Tu), (20)

where Tu and Tb are the temperatures of the unburnt and burnt mixture. Fordefiniteness, we take Tu = 300K and Tb = 2, 100K, although it is only the ratioTb/Tu = 7 that is significant. Consistent with the ideal gas law, the density andspecific volume are then specified as

ρ(T ) = ρuTuT, v(T ) = vu

T

Tu, (21)

where ρu = 1/vu are the unburnt values.The molecular diffusivity is specified as

D(T ) = Du

(T

Tu

)1.72

, (22)

which is based on an accurate approximation for methane combustion[16], with Du

being the unburnt value.For each quantity introduced, the value in the burnt stream is denoted by ρb, vb

and Db. The density and specific volume ratios are 7 (like the temperature), whilethe diffusivity ratio is Db/Du ≈ 28.The chemical source term is specified to be

S(φ) = 0 for φ < φR (23)

=1

τc

4

27z(1− z)2 for φ ≥ φR,

where τc is the chemical time scale, φR is taken to be 0.4, and z is defined by

z ≡1− φ

1− φR. (24)

The resulting normalized function S(φ)τc is shown in Fig. 1. The factor of 4/27 inEq. 23 is chosen to make the maximum value of S(φ)τc unity.For this simple 1D case, it is not necessary to consider the momentum equation,

as the velocity is determined by mass conservation. Specifically, in the frame fixedwith the flame, the mass flux is uniform, so that the velocity U is obtained fromthe relation

ρU = ρuuF . (25)

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φ0 0.2 0.4 0.6 0.8 1

S(φ)τ

c

0

0.2

0.4

0.6

0.8

1

Figure 1. Normalized reaction source term, Eq. 23.

We arbitrarily set ρu, Du and τc to unity, and so all quantities obtained can beconsidered to be normalized by them.

3.2 Solutions considered

For this test case, we consider the following five solutions:

(1) The numerically-accurate solution of the governing equation (Eq. 3), i.e.,the laminar-flame solution; or, equivalently, the DNS solution.

(2) The numerically-accurate solution of the LES/PDF random-walk equationsin the DNS limit.

(3) The numerically-accurate solution of the LES/PDF mean-drift equationsin the DNS limit.

(4) The numerical solution of the LES/PDF random-walk equations in the DNSlimit.

(5) The numerical solution of the LES/PDF mean-drift equations in the DNSlimit.

3.3 DNS solution

The laminar-flame or DNS solution is obtained by solving Eq. 3 for the case con-sidered by an accurate finite-difference method. The resulting composition profileis shown in Fig. 2. In general, the flame speed is denoted by uF and the flamethickness (defined below) is denoted by δF . However, for this reference, DNS solu-tion, these quantities are denoted by sL and δL, i.e., the laminar flame speed andthickness.The measure of flame thickness we use is the secant thickness, defined as follows.

Let x1/4 and x3/4 denote the locations at which the composition φ has the values1/4 and 3/4, respectively. Then we define the thickness as:

δF = 2(x3/4 − x1/4). (26)

As illustrated in Fig. 2, the geometric interpretation of this thickness is that it isthe distance between the intersections of the secant through x1/4 and x3/4 and the

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x/δL-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

φ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2. Laminar flame profile. The symbols are the points on the profile where φ = [1/4 3/4], and thedashed line through them is the secant used to define the flame thickness.

pure stream values, φ = 0 and φ = 1. We also define the center of the flame to beat the mid-point between x1/4 and x3/4. Thus, when the flame is centered at theorigin (x = 0), it follows that x1/4/δF = −1/4 and x3/4/δF = 1/4 .The (normalized) laminar flame speed and thickness are found to be

sL/(Du/τc)1/2 = 0.81 and δL/(Duτc)

1/2 = 7.6.

3.4 Numerically-accurate solution for the mean-drift model

As mentioned in Sec. 2.6, the numerically-accurate solution of the LES/PDF mean-

drift model equations are consistent with the DNS limit. Thus the solution for φ isjust the laminar flame solution, and the variance of φ is zero. The value specifiedfor the mixing rate Ω is immaterial.

3.5 Numerically-accurate solution for the random-walk model

In contrast, the random-walk model is not consistent with the DNS limit. In thiscase the solution depends on the specified (normalized) mixing rate Ωτc.Numerically-accurate solutions for the random-walk model are obtained using

the mesh-free particle method described in Appendix A. Consistent with Eq. 9,the mixing rate is taken to be proportional to the molecular diffusivity, i.e.,

Ω = ΩuD

Du, (27)

with the unburnt mixing rate, Ωu, being a specified parameter.The computed normalized flame speed and thickness are shown in Figs. 3 and

4 as functions of the normalized unburnt mixing rate, Ωuτc. As may be seen fromthese figures, as Ωuτc increases beyond unity, the flame speed and thickness becomeclose to their laminar values.Since the composition is non-reactive for φ < φR = 0.4, only by mixing does

the composition rise from zero to φR. As a consequence, for small Ωuτc, mixingis the rate-limiting process. If one assumes that (for small Ωuτc), the flame speed

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Ωuτc

10-2

10-1

100

101

uF/s

L

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 3. For the random-walk model, the normalized flame speed as a function of the normalized mixingrate. Dashed line: 3(Ωuτc)1/2.

Ωuτc

10-2

10-1

100

101

δF/δ

L

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure 4. For the random-walk model, the normalized flame thickness as a function of the normalizedmixing rate. Dashed line: 3.7(Ωuτc)−1/2.

and thickness are determined by D and Ω, independent of τc, then dimensionalarguments dictate that the uF scales with (DuΩu)

1/2, and that δF scales with(Du/Ωu)

1/2. As may be seen from Figs. 3 and 4, the results for the 4 smallestvalues of Ωuτc investigated are consistent with these scalings.It is interesting to observe that, as Ωuτc decreases, uF /sL first increases and

attains a maximum of about 1.3 around Ωuτc ≈ 0.3, before decreasing, eventuallywith the mixing-limited scaling.Figure 5 shows σφ,max, the maximum value (over all x) of the standard deviation

of the composition. For small Ωuτc, the standard deviation has the quite large valuearound 0.4, independent of τc. For large Ωuτc, it decreases as (Ωuτc)

−1/2, consistentwith the balance of production and dissipation.Figure 6 shows a scatter plot of particle composition through the flame. For

the case shown (Ωuτc = 0.01), mixing is rate limiting, and reaction in comparisonis very fast. As a consequence, there are very few partially-burnt particles (i.e.,

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Ωuτc

10-2

10-1

100

101

σφ,m

ax

10-2

10-1

100

Figure 5. For the random-walk model, the maximum standard deviation of composition as a function ofthe normalized mixing rate. Dashed line: 0.14(Ωuτc)−1/2.

x/δL-6 -4 -2 0 2 4 6 8

φ∗,φ

0

0.2

0.4

0.6

0.8

1

Figure 6. For the random-walk model with Ωuτc = 0.01, a scatter plot of particle composition φ∗ against

position. Solid line: resolved composition φ. Dashed line: laminar flame profile.

0.4 < φ∗ < 1), and many fully-burnt particles (i.e., φ∗ = 1).In summary, the numerically-accurate solution to the random-walk model in

the DNS limit depends on the normalized mixing rate Ωuτc. To the extent thatthe residual variance is non-zero, the model is inconsistent with the DNS limit.However, for Ωuτc greater than unity, the flame speed and flame thickness areclose to the laminar values, and the residual standard deviation is less than 10%.Consequently, in spite of the spurious residual variance, the random-walk modelmay be considered to be a useful and accurate model provided that the mixingrate is sufficiently large.

3.6 Cloud-in-cell solutions to the model equations

In practice, the LES/PDF equations are not solved accurately (as considered inthe previous sub-section), but instead are solved by a numerical method that may

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incur significant error. This is because the mesh spacing h is typically chosen not tobe very small compared to the resolution length scale ∆, and indeed may be largerthan the smallest scale in the accurate solution to the LES/PDF equations (e.g.,the flame thickness δF ). We therefore consider here the numerical solution of theLES/PDF equations using the random-walk and mean-drift models obtained usinga cloud-in-cell particle/mesh method. This is the type of method typically used inpractice. The details of the current implementation are given in Appendix B.For the given test case being considered, the numerical solution to the LES/PDF

equations depends on three quantities: the specified resolution scale ∆, the specifiedmixing-model constant Cm, and the mesh spacing h. (We take the number ofparticles N to be large enough, and the time step ∆t to be small enough, such thatonly the spatial discretization error is significant.) It is convenient to reduce thenumber of parameters that need consider to two, which is possible since Cm and ∆enter solely through the combination Cm/∆

2 (in Eq. 9). To this end, we re-expressthe mixing rate as

Ω = CDD

h2, (28)

where comparison with Eq. 9 shows that CD and Cm are related by

CD = Cm

(h

∆

)2

. (29)

We then take the two parameters to be considered to be CD and h/δL. Note that,for example, the value CD = 1 corresponds both to Cm = 1, h/∆ = 1, and toCm = 4, h/∆ = 1/2, etc.Figure 7 shows the computed flame speed as a function of the normalized mesh

spacing for both models and for two values of the model coefficient CD. For smallh/δL both models attain the same asymptote, but with uF /sL being greater thanunity, specifically 1.2 for CD = 1 and 1.6 for CD = 4. As with the accurate solution,with increasing mesh spacing, the flame speed of the random-walk model achieves amaximum, before decreasing. On the other hand the flame speed of the mean-driftmodel decreases monotonically, and is similar to that of the random walk modelfor large h/δL.The key to understanding many of these observations is to appreciate that the

CIC implementation of the IEM model incurs a numerical smearing error. A simpleanalysis shows that, to first order, this error is equivalent to there being an addi-tional numerical diffusivity Dn, which is proportional to Ωh2. With α denoting thecoefficient of proportionality, we have

Dn = αΩh2 = αCDD. (30)

Note that, for fixed CD, this numerical diffusivity does not vanish as h/δL tendsto zero, and it increases with CD. This explains the observed values of uF /sL (forsmall h/δL) being greater than unity, and larger for CD = 4 than for CD = 1.(Below, we empirically determine the value α = 0.4.)This reasoning can be made quantitative as follows. We define the augmented

diffusivity to be

Da ≡ D +Dn = D(1 + αCD). (31)

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h/δL10

-110

010

1

uF/s

L

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

RW, CD = 1

RW, CD = 4

MD, CD = 1

MD, CD = 4

Figure 7. Using the cloud-in-cell method, the normalized flame speed as a function of the normalizedmesh spacing for (from bottom to top at the right): RW model, CD = 1; MD model, CD = 1; RW model,CD = 4; MD model, CD = 4.

Just as the laminar flame speed and thickness scale as (Du/τc)1/2 and (Duτc)

1/2,respectively, the numerical flame speed and thickness can be expected to scale as(Da/τc)

1/2 and (Daτc)1/2, respectively, and hence are larger than the laminar values

by a factor of

(Da

Du

)1/2

= (1 + αCD)1/2. (32)

We thus define the augmented laminar flame speed and thickness to be

sa ≡ (1 + αCD)1/2 sL, (33)

and

δa ≡ (1 + αCD)1/2 δL, (34)

and, on Fig. 8, we re-plot the computed flame speeds, but now with normalizationby sa and δa. As may be seen, with the empirically-determined value of α = 0.4,this scaling is successful in accurately yielding uF = sa for small h/δa for all fourcases.Figure 9 shows the computed flame thicknesses normalized by δa. As may be

seen, in all four cases, δF converges to δa as the mesh is refined. For large h, theflame thickness scales essentially with h, with δF ≈ 4h for the random-walk modelwith CD = 1, and δF ≈ h for the mean-drift model with CD = 4.In the above we consider CD to be a fixed parameter. If instead we revert to

considering Cm to be fixed, then from Eq. 33 we obtain (for small h/δL and h/∆)

sasL

=

(1 + αCm

[h

∆

]2)1/2

(35)

≈ 1 +1

2αCm

[h

∆

]2,

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h/δa10

-210

-110

010

1

uF/s

a

0

0.2

0.4

0.6

0.8

1

1.2

RW, CD = 1

RW, CD = 4

MD, CD = 1

MD, CD = 4

Figure 8. Using the cloud-in-cell method, the flame speed normalized by sa as a function of the meshspacing normalized by δa for (from bottom to top at the right): RW model, CD = 1; RW model, CD = 4;MD model, CD = 1; MD model, CD = 4.

h/δa10

-210

-110

010

1

δF/δa

100

101

RW, CD = 1

RW, CD = 4

MD, CD = 1

MD, CD = 4

Figure 9. Using the cloud-in-cell method, the flame thickness as a function of the mesh spacing, bothnormalized by δa for (from bottom to top at the right): MD model, CD = 4; MD model, CD = 1; RWmodel, CD = 4; RW model, CD = 1. The lower and upper dashed lines are δF = h and δF = 4h,respectively.

and similarly for δa/δL. The important conclusions are that, for fixed Cm, thecorrect laminar flame speed and thickness are obtained as the grid spacing h/δLtends to zero when ∆/δL is fixed, but not when h/∆ is fixed.

3.7 Matching the laminar flame speed

It is natural to ask: with the CIC implementation of the LES/PDF models, isit possible to specify the mixing rate Ω so that the flame speed uF matches thelaminar flame speed sL, even if the mesh size h is not small compared to the laminarflame thickness δL? The answer is: yes.First consider the case when h/δL is large. In this case: the numerical diffusion

Dn = αΩh2 dominates the molecular diffusion; the flame thickness scales with h

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h/δL10

-210

-110

010

1

CL

10-1

100

Figure 10. Empirical coefficients CL(h/δL) (Eqs. 37 and 38) yielding flame speeds equal to the laminarflame speed for the CIC implementation of the random-walk model (lower curve at left) and of the mean-drift model (upper curve on left).

(see Fig. 9); and reaction is not rate limiting. Hence the flame speed is determineddominantly by Dn and h, with D and τc not being significant. It follows (on dimen-sional grounds) that the flame speed scales as hΩ, and hence, to match the laminarflame speed (which scales as D/δL), Ω has to be specified to scale as D/(hδL).Thus, for large h/δL, the appropriate specification is

Ω = CLD

hδL, (36)

where CL is a constant. More generally, we can consider CL to be a coefficientdependent on h/δL, which tends to a constant for large h/δL.Before proceeding, we make the following observation based on the flame speed

scaling with Ωh for large h/δL. It follows from Eq. 36 that, by construction, for fixedCL, uF /sL is independent of h/δL. It further follows from the relations betweenCL, CD and Cm that uF /sL varies as (h/δL)

−1 for fixed CD, and as hδL/∆2 =

(h/∆)2(h/δL)−1 for fixed Cm.

We determine empirically that the function CL(h/δL) which yields uF = sL iswell approximated for the random-walk model by

CL = 3.3

[1− exp

(−2

h

δL

)], (37)

and for the mean-drift model by

CL = min

(2.5, 3.5

[h

δL

]1/2). (38)

These functions are shown in Fig. 10.Note that, with everything except h fixed: for fixed CD, Ω varies as h−2; for large

h/δL the above expressions for CL imply Ω varying as h−1; while for small h/δL,CL for the RW model (Eq. 37) implies Ω varying as h0, whereas CL for the MDmodel (Eq. 38) implies Ω varying as h−1/2.

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h/δL10

-110

010

1

uF/s

L

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

RW

MD

Figure 11. Normalized flame speed against normalized mesh spacing for the CIC implementations of therandom-walk and mean-drift models with the mixing rate specified by Eqs. 36–38.

The computed flame speeds using Ω given by Eqs. 36–38 are shown in Fig. 11.As may be seen, with these specifications of CL, both models do indeed yield flamespeeds close to the laminar flame speed.While both models yield the same flame speed, their solutions are quite differ-

ent. While the mean-drift model is consistent with the DNS limit in producingno residual fluctuations, the random-walk model has large residual fluctuations(around 30%), even for small h/δL. This is because, with the RW model with CL

given by Eq. 37, as h/δL tends to zero, Ω tends to the value 6.6D/δ2L, independentof h; whereas, in order to suppress residual fluctuations and hence to reproduce theDNS limit, Ω must vary as a negative power of h/δL (as it does for a fixed valueof CD).As a further manifestation of the differences between the model solutions, Fig. 12

shows the normalized flame thicknesses. For small h/δL, the mean-drift modelcorrectly yields δF = δL; and for large h/δL it yields δF ≈ h. In contrast, therandom-walk model yields consistently thicker flames, with δF ≈ 2δL for smallh/δL.We conclude this sub-section with the following caveats and reservations. A great

virtue of PDF methods is that they are generally applicable to all modes andregimes of combustion. However, the specification of the mixing rate Ω by Eqs. 36–38 is anything but general. It is particular to homogeneously-premixed combustion,so that a unique laminar flame thickness δL can be defined. Furthermore, the ap-propriate specification of the coefficient CL depends on the details of the numericalimplementation.

4. Strained Methane/Air Premixed Laminar Flame

4.1 Definition of the test case

The flow considered consists of two opposed round jets emanating from nozzlesof diameter D = 12.7 mm placed at a distance of d = 16 mm apart. One ofthe nozzles emits a fresh (unburnt) mixture of CH4/O2/N2 at equivalence ratioφu = 0.85 (O2/N2 = 30/70 molar ratio), temperature Tu = 294K and pressure

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h/δL10

-110

010

1

δF/δL

100

101

RW

MD

Figure 12. Normalized flame thickness against normalized mesh spacing for the CIC implementations ofthe random-walk model using Eq. 37 (upper curve) and the mean-drift model using Eq. 38 (lower curve).The dashed line is δF = h.

1 atm. The other nozzle emits fully-burnt combustion products at temperatureTad = 2, 435K, corresponding to the chemical-equilibrium mixture with the sameenthalpy, elemental composition and pressure as the reactant stream. The axialvelocity of the reactant stream is Uu = 1 m/s, and that of the product stream istaken to be twice this value so that the resulting laminar flame is approximatelymid-way between the nozzles. The nominal mean strain rate is thus 2Uu/d = 125s−1. The radial velocities at the nozzle exits are taken to be zero.The axial coordinate is denoted by z, with z = −8 mm being the product-stream

nozzle exit, and z = 8 mm being the reactant-stream nozzle exit.The methane combustion is described by 16-species augmented reduced mech-

anism for methane oxidation[24]. The thermal diffusivity is obtained from theCHEMKIN transport library, and then the (equal) species diffusivities are obtainedby taking the Lewis numbers to be unity.

4.2 OPPDIF calculations

As is well known[25], for the flow considered, the profiles along the centerline canbe obtained from the solution of 1D equations. These equations are solved usingthe OPPDIF code within CHEMKIN-PRO[21]. Figure 13(a) shows the computedaxial profiles of the axial velocity and temperature. Figure 13(b) shows the profileof ρU/ρu. The value of the laminar flame speed sL is taken to be the value of ρU/ρuat the location of maximum temperature gradient. This location is around z = 4mm, and the resulting value of sL is 55.8 cm/s. The secant thickness (obtainedfrom the normalized temperature profile (T − Tu)/(Tad − Tu) and Eq. 26) is δL =0.334 mm.

4.3 LES/PDF calculations

LES/PDF calculations are performed in the DNS limit using the coupledNGA/HPDF code[6, 26, 27]. The number of grid points in the radial and az-imuthal directions are kept constant at 48 and 4 respectively. while the number ofgrid points in the axial direction, nx, is varied from 12 to 384 in order to span the

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−8 −6 −4 −2 0 2 4 6 8−2

−1

0

1

2

3

4

5

U(m

/s)

z (mm)−8 −6 −4 −2 0 2 4 6 8

0

500

1000

1500

2000

2500

T(K

)

(a)

−8 −6 −4 −2 0 2 4 6 8−1

−0.5

0

0.5

1

z (mm)

ρU/

ρu

(m/s)

(b)

Figure 13. Laminar profiles from the OPPDIF calculations of (a) velocity and temperature and (b) ρU/ρuas a function of axial coordinate between the two nozzles.

Axial (mm)

Radial(m

m)

2 4 6 8 10 12 14 160

5

10

15

20

25

∆/δL

1

2

3

4

5

Figure 14. Contour plot of normalized filter-width from previous 3D LES/PDF simulations of turbulentpremixed counterflow flames[28].

range of values of ∆/δL shown in Fig. 14. This range is observed in our previous3D LES/PDF simulations of turbulent premixed counterflow flame[28]. Note thatthe filter-width in the present simulations is taken to be equal to the (uniform)grid spacing in the axial direction, i.e. ∆ = h.The results reported below depend on: the model used (random walk or mean

drift); the value of the mixing-rate constant Cm; and, the specified axial grid spacingand resolution length scale h = ∆. The values of Cm used are 1.0 and 4.0. By usingdifferent grids, h/δL is varied between 0.13 and 4.0.

4.4 Results

Figures 15 and 17 show the normalized flame speed and flame thickness as func-tions of the normalized grid spacing for both models and for both values of Cm

investigated. The flame speed uF and the flame thickness δF are obtained from the

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10−1

100

101

10−2

10−1

100

h/

δL

uF/s L

Cm = 4.0, RWCm = 4.0,MDCm = 1.0, RWCm = 1.0,MD

Figure 15. Normalized flame speed as a function of normalized grid spacing for: Cm = 4.0, RW (bluedot); Cm = 4.0, MD (blue star); Cm = 1.0, RW (red dot); Cm = 1.0, MD (red star).

10−3

10−2

0.8

1

1.2

1.4

∆t/τc

uF/s L

Cm = 4.0,MD, h/δL = 0.25Cm = 1.0,MD, h/δL = 0.25

Figure 16. Normalized flame speed as a function of normalized time step for Cm = 4.0, MD, h/δL =0.25 (blue dot) and Cm = 1.0, MD, h/δL = 0.25 (red dot). Here, τc is defined by τc ≡ δL/sL.

centerline profiles of the resolved velocity, density and temperature in the sameway as for the OPPDIF calculations (as described in Sec. 4.2).For the flame speed, the observed behavior (Fig. 15) is broadly the same as for

the CIC solution for the idealized, unstrained 1D flame considered in Sec. 3.6 (seeFig. 7). For the random-walk model, uF /sL has an asymptote, larger than unity,for small h/δL, and then decreases with increasing h/δL. The values of uF /sLfor Cm = 4 are consistently higher than those for Cm = 1. The only qualitativedifference compared to the results in Fig. 7 is that a local maximum in uF /sLaround h/δL ≈ 0.4 is not observed.For the mean-drift model, the results are qualitatively similar. The only point

of note is that, for Cm = 1, the asymptotic value of uF /sL is about 0.9, i.e., lessthan unity. Further investigation reveals that this is due to time-stepping error. Asshown in Fig. 16, as the time step decreases, uF /sL increases, plausibly exceedingunity for smaller time steps than investigated.

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10−1

100

101

100

101

h/

δL

δF

/

δL

Cm = 4.0, RWCm = 4.0,MDCm = 1.0, RWCm = 1.0,MD

Figure 17. Normalized flame thickness as a function of normalized grid spacing for: Cm = 4.0, RW (bluedot); Cm = 4.0, MD (blue star); Cm = 1.0, RW (red dot); Cm = 1.0, MD (red star). The dashed line isδF = 2.5h.

Figure 17 shows the normalized flame thickness as a function of the normalizedgrid spacing. The observations are the same as for the CIC solutions for the ideal-ized, unstrained flame (see Fig. 9). The flame thickness is generally larger for therandom-walk model than for the mean-drift model, and decreases with increasingCm. On the finest grids, δF /δL approaches an asymptote greater than unity; whileon coarse grids the flame thickness is approximately δF ≈ 2.5h (in three out of thefour cases).Figure 18 shows scatter plots on the centerline of the particle temperature vs.

position, color coded by the mass fraction of OH. The left-hand column is for therandom-walk model, and the right-hand column for the mean-drift model. The toptwo rows are for Cm = 4, and the bottom two for Cm = 1. The first and third rowsare for the relatively coarse grid (h/δ = 1.4), while the second and fourth rows arefor the finer grid (h/δ = 0.5). The symbols on the top axis of each plot show thelocations of the grid nodes.The first observation to be made from these scatter plots is that the implemen-

tation of the mean-drift model is successful in producing negligible residual fluctu-ations in all cases. It may also be observed that the particles are, to some extent,able to resolve the flame profile below the grid scale. For the random-walk modelthere is appreciable scatter (inconsistent with the DNS limit), which decreases withincreasing Cm and with decreasing h/δL.The results presented in this section serve to confirm the validity of the model

implementations in the 3D LES/PDF code in the DNS limit. They also confirmthat the results obtained for the idealized, unstrained flame using the 1D MAT-LAB script are representative of those obtained with the much more complex 3DLES/PDF code.

5. Discussion

Based on the results presented above and other considerations, it is clear thatthe mean-drift model has significant theoretical advantages over the random-walkmodel. These advantages are:

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−1 0 1 2 30

500

1000

1500

2000

2500

z (mm)

T∗

Y∗

OH

0.000

0.002

0.004

0.006

0.008

0.010

0.012

(a)

−1 0 1 2 3 40

500

1000

1500

2000

2500

z (mm)

T∗

Y∗

OH

0.002

0.004

0.006

0.008

0.010

0.012

(b)

1 2 3 40

500

1000

1500

2000

2500

z (mm)

T∗

Y∗

OH

0.000

0.002

0.004

0.006

0.008

0.010

0.012

(c)

5 6 7 80

500

1000

1500

2000

2500

z (mm)

T∗

Y∗

OH

0.002

0.004

0.006

0.008

0.010

(d)

−3 −2 −1 0 1 20

500

1000

1500

2000

2500

z (mm)

T∗

Y∗

OH

0.000

0.002

0.004

0.006

0.008

0.010

0.012

(e)

−2 −1 0 10

500

1000

1500

2000

2500

z (mm)

T∗

Y∗

OH

0.002

0.004

0.006

0.008

0.010

0.012

(f)

3 4 5 60

500

1000

1500

2000

2500

z (mm)

T∗

Y∗

OH

0.000

0.002

0.004

0.006

0.008

0.010

0.012

(g)

0 1 2 30

500

1000

1500

2000

2500

z (mm)

T∗

Y∗

OH

0.002

0.004

0.006

0.008

0.010

(h)

Figure 18. Scatter plots of particle temperature on the centerline color-coded by particle OH species massfraction for (a) Cm = 4.0, RW, h/δL = 1.4, (b) Cm = 4.0, MD, h/δL = 1.4, (c) Cm = 4.0, RW, h/δL = 0.5,(d) Cm = 4.0, MD, h/δL = 0.5, (e) Cm = 1.0, RW, h/δL = 1.4, (f) Cm = 1.0, MD, h/δL = 1.4, (g) Cm

= 1.0, RW, h/δL = 0.5, (h) Cm = 1.0, MD, h/δL = 0.5. The grid points are shown as black circles at thetop of the figures.

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(1) Differential diffusion can be implemented in the mean-drift model, whereaswith the random-walk model the unity-Lewis-number assumption is un-avoidable.

(2) The mean-drift model is consistent with the DNS limit, whereas therandom-walk model has spurious production of variance leading to signifi-cant residual fluctuations.

(3) In the CIC implementation of the models, on a given mesh, the flame thick-ness δF given by the mean-drift model is significantly smaller (i.e., closerto δL) than that given by the random-walk model.

However, the mean-drift model also has distinct disadvantages in ease of imple-mentation and computational cost. Specifically:

(1) With the mean-drift model, special measures must be taken to guarantee re-alizability (e.g., to ensure that species mass fractions remain non-negative).

(2) The mean-drift model requires the evaluation of the nφ resolved composi-

tion fields φ(x, t) (which is computationally expensive) and the solution ofnφ PDEs to determine the mean drift.

In contrast, the random-walk model can be implemented in an efficient, mesh-free algorithm that automatically guarantees realizability and does not require theevaluation of the resolved compositions. Because of these practical advantages,and in spite of its theoretical disadvantages, we do not advocate abandoning therandom-walk model: it may be adequate (and simpler and cheaper) for a range ofproblems in which differential diffusion and consistency with the DNS limit are notcrucial.

6. Conclusions

The principal conclusions to be drawn from this work are as follows.

(1) In considering LES in the DNS limit, it is advantageous to define the re-solved quantities as conditional means rather than as filtered fields. Withthe conditional-mean approach, in the DNS limit, the resolved fields in LESare just the DNS fields, and there are no residual fluctuations. In contrast,with the filtering approach, the resolved fields differ from the DNS fields (byof order (∆/δ)2), and the residual variances are non-zero (and are, again,of order (∆/δ)2).

(2) The behavior of LES/PDF models in the DNS limit has been studied byapplying them to two laminar premixed flames: an idealized unstrainedflame, and a strained methane/air flame.

(3) The numerically-accurate solutions to the LES/PDF equations incorporat-ing the mean-drift model are consistent with the DNS limit.

(4) A mesh-free particle method has been implemented to obtain numerically-accurate solutions to the LES/PDF equations incorporating the random-walk model.

(5) These numerically-accurate solutions to the LES/PDF equations incorpo-rating the random-walk model are not consistent with the DNS limit, inthat the residual fluctuations are non-zero. However, if the mixing rate Ωis sufficiently large (e.g., Ωτc & 1), then the flame speed and thickness areclose to the laminar values, and the residual fluctuations are not large (e.g.,less than 10%).

(6) The cloud-in-cell implementation of the IEM mixing model incurs a smear-

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REFERENCES 25

ing error which is equivalent to a numerical diffusion Dn which scales asΩh2.

(7) With the standard specification of the mixing rate, with a constant valueof Cm, as the mesh is refined (i.e., h/δL → 0), the LES/PDF solutions tendto the DNS limit if the resolution scale is fixed (i.e., ∆/δL is fixed), but notif h/∆ is fixed.

(8) On coarse meshes (h/δL ≫ 1): the flame thickness δF scales with the meshspacing h; the mixing rate Ω and the mesh spacing h are the controllingparameters (with the molecular diffusivity D and the reaction time scale τcnot being controlling); and hence the flame speed uF scales as Ωh.

(9) On coarse meshes, and with Cm and h/∆ fixed, the normalized flame speeduF /sL scales with (h/δL)

−1.(10) With both RW and MD models, it is possible to specify the mixing rate (by

Eqs. 36–38) so that the flame speed uF matches the laminar flame speed sL,even on coarse meshes. It is emphasized that this is a non-general model.

(11) Computations have been performed of a strained, premixed, methane/air,opposed-jet flame, using the same 3D code used in previous LES/PDFstudies. The results confirm the validity of the model implementations inthe 3D code in the DNS limit; and they confirm that the results obtainedfor the idealized, unstrained flame are representative of those obtained withthe much more complex 3D LES/PDF code.

(12) Whereas the mean-drift has theoretical advantages (in allowing differentialdiffusion to be implemented and being consistent with the DNS limit), therandom-walk has the practical advantages of being simpler to implementand being computationally less expensive.

Acknowledgements

This material is based upon work supported by the U.S. Department of EnergyOffice of Science, Office of Basic Energy Sciences under Award Number DE-FG02-90ER14128.

References

[1] S.B. Pope, PDF methods for turbulent reactive flows, Prog. Energy Combust. Sci. 11 (1985),pp. 119–192.

[2] D.C. Haworth, Progress in probability density function methods for turbulent reacting flows,Prog. Energy Combust. Sci. 36 (2010), pp. 168–259.

[3] S.B. Pope, Small scales, many species and the manifold challenges of turbulent combustion,Proc. Combust. Inst. 34 (2013), pp. 1–31.

[4] F.A. Jaberi, P.J. Colucci, S. James, P. Givi, and S.B. Pope, Filtered mass density functionfor large-eddy simulation of turbulent reacting flows, J. Fluid Mech. 401 (1999), pp. 85–121.

[5] V. Raman and H. Pitsch, A consistent LES/filtered-density function formulation for thesimulation of turbulent flames with detailed chemistry, Proc. Combust. Inst. 31 (2007), pp.1711–1719.

[6] Y. Yang, H. Wang, S.B. Pope, and J.H. Chen, Large-eddy simulation/probability densityfunction modeling of a non-premixed CO/H2 temporally evolving jet flame, Proc. Combust.Inst. 34 (2013), pp. 1241–1249.

[7] I.A. Dodoulas and S. Navarro-Martinez, Large eddy simulation of premixed turbulent flamesusing the probability density function approach, Flow, Turbul. Combust. 90 (2013), pp. 645–678, Available at http://dx.doi.org/10.1007/s10494-013-9446-z.

[8] J. Kim and S.B. Pope, Effects of combined dimension reduction and tabulation on the simula-

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26 REFERENCES

tions of a turbulent premixed flame using large-eddy simulation/probability density function,Combust. Theory Modelling 18 (2014), pp. 388–413.

[9] S.B. Pope and R. Tirunagari, Advances in probability density function methods for turbulentreactive flows, in Proceedings of the Nineteenth Australasian Fluid Mechanics Conference,RMIT University, Melbourne, 2014.

[10] S.B. Pope, Turbulent Flows, Cambridge University Press, Cambridge, 2000.[11] H. Pitsch, Large-eddy simulation of turbulent combustion, Annu. Rev. Fluid. Mech. 38 (2006),

pp. 453–482.[12] S.B. Pope, Self-conditioned fields for large-eddy simulations of turbulent flows, J. Fluid Mech.

652 (2010), pp. 139–169.[13] R.S. Rogallo and P. Moin, Numerical simulation of turbulent flows, Annu. Rev. Fluid. Mech.

17 (1985), pp. 99–137.[14] R.O. Fox, Computational Models for Turbulent Reactive Flows, Cambridge University Press,

New York, 2003.[15] S.B. Pope, Ten questions concerning the large-eddy simulation of turbulent flows, New J.

Phys. 6 (2004), p. 35.[16] K.A. Kemenov and S.B. Pope, Molecular diffusion effects in LES of a piloted methane-air

flame, Combust. Flame 157 (2011), pp. 240–254.[17] R.S. Barlow and J.H. Frank, Effects of turbulence on species mass fraction in methane/air

jet flames, Proc. Combust. Inst. 27 (1998), pp. 1087–1095.[18] O.T. Stein, B. Boehm, A. Dreizler, and A.M. Kempf, Highly-resolved LES and PIV analysis

of isothermal turbulent opposed jets for combustion applications, Flow, Turbul. Combust. 87(2011), pp. 425–447.

[19] A.M. Ruiz, G. Lacaze, and J.C. Oefelein, Flow topologies and turbulence scales in a jet-in-cross-flow, Phys. Fluids 27 (2015), p. 045101.

[20] J. Villermaux and J.C. Devillon, Representation de la coalescence et de la redispersion desdomaines de segregation dans un fluide par un modele d’interaction phenomenologique, inProceedings of the 2nd International Symposium on Chemical Reaction Engineering, Elsevier,New York, 1972, pp. 1–13.

[21] R.J. Kee, F.M. Rupley, J.A. Miller, M.E. Coltrin, J.F. Grcar, E. Meeks, H.K. Moffat, A.E.Lutz, G. Dixon-Lewis, M.D. Smooke, J. Warnatz, G.H. Evans, R.S. Larson, R.E. Mitchell,L.R. Petzold, W.C. Reynolds, M. Caracotsios, W.E. Stewart, P. Glarborg, C. Wang, and O.Adigun, CHEMKIN Collection, Release 3.6, in Reaction Design, Inc., San Diego, CA, 2000.

[22] R. McDermott and S.B. Pope, A particle formulation for treating differential diffusion infiltered density function methods, J. Comput. Phys. 226 (2007), pp. 947–993.

[23] S. Viswanathan, H. Wang, and S.B. Pope, Numerical implementation of mixing and moleculartransport in LES/PDF studies of turbulent reacting flows, J. Comput. Phys. 230 (2011), pp.6916–6957.

[24] C.J. Sung, C.K. Law, and J.Y. Chen, An augmented reduced mechanism for methane oxidationwith comprehensive global parametric validation, Proc. Combust. Inst. 27 (1998), pp. 295–304.

[25] R.J. Kee, M.E. Coltrin, and P. Glarborg, Chemically reacting flow: theory and practice, Wiley,Hoboken, 2003.

[26] O. Desjardins, G. Blanquart, G. Balarac, and H. Pitsch, High order conservative finite dif-ference scheme for variable density low Mach number turbulent flows, J. Comput. Phys. 227(2008), pp. 7125–7159.

[27] H. Wang and S.B. Pope, Large eddy simulation/probability density function modeling of aturbulent CH4/H2/N2 jet flame, Proc. Combust. Inst. 33 (2011), pp. 1319–1330.

[28] R. Tirunagari and S.B. Pope, Computational study of turbulent premixed counterflow flames,in 9th U.S. National Combustion Meeting, Cincinnati, Ohio, 2015.

[29] S.B. Pope, A model for turbulent mixing based on shadow-position conditioning, Phys. Fluids25 (2013), p. 110803.

[30] S.B. Pope and R. Gadh, Fitting noisy data using cross-validated cubic smoothing splines,Commun. Stat. Simul. Comput. 17 (1988), pp. 349–376.

[31] M. Muradoglu, S.B. Pope, and D.A. Caughey, The hybrid method for the PDF equations ofturbulent reactive flows: consistency conditions and correction algorithms, J. Comput. Phys.172 (2001), pp. 841–878.

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Appendix A. Mesh-Free Particle Method

A mesh-free particle method is used to obtain numerically-accurate solutions tothe LES/PDF equations for the random-walk model in the DNS limit appliedto the idealized, unstrained, 1D, plane laminar flame. The equations are solvedin the frame in which the reactants are at rest at x = −∞. From a specifiedinitial condition, the solution is advanced in time through small time steps ∆t,and eventually reaches a fully-developed state, with the flame moving to the leftat a constant speed uF . The solution domain extends from pure reactants to pureproducts, and is moved with the flame.There are a large number N of particles, the nth having position X(n)(t) and

composition φ(n)(t) at time t. The solution is advanced from time t to time t+∆tthrough the following sequence of operations (with some details to follow). Prior tothe start of each step, the particles are sorted in position so that X(n) ≥ X(n−1).

(1) The particle diffusivities are evaluated, and the mean diffusivity at the par-

ticle locations, D(n), is estimated using a cross-validated smoothing linearspline.

(2) The mixing rate Ω(n) is determined from D(n).(3) The mixing sub-step is performed using the near-neighbor implementation

of the IEM model[29].(4) The reaction sub-step is performed.(5) The particles are moved by the diffusion term of the random walk, and then

re-sorted.(6) The particles are moved by the drift term in the random walk.(7) As necessary, the domain is moved, and particles are removed and added

as necessary at the boundaries.

In order to estimate means, the particles are partitioned into odd and even-numbered particles. Linear smoothing splines[30] (with smoothing parameter s)are generated based on the two sets of particles, and the cross-validation error χ(s)is computed. An iteration is performed to determine the value of the smoothingparameter that minimizes the error. (For computational efficiency, if there are alarge number of particles, then this procedure is applied, not to individual particles,but to clusters of adjacent particles. Here the number of clusters is limited to 1,000.)The particles move due to the drift term in the random walk (Eq. 7) according to

dX∗/dt = [U+v∇(〈ρ〉D)]∗. Fortunately, however, this drift coefficient does not needto be evaluated, because the effect of the drift can be implemented implicitly byenforcing a consistency condition[31]. Every particle is ascribed the same mass perunit cross-sectional area, m′′. The nth particle, with specific volume v(n), thereforehas a length associated with it of ℓ(n) = m′′v(n). The consistency condition is thatthe particle length is equal to the geometric length that the particle occupies. Thiscondition is strongly enforced by (at the end of the time step) setting the particlepositions by

X(n+1) = X(n) +1

2[ℓ(n+1) + ℓ(n)]. (A1)

The solutions reported in Sec. 3.5 are performed with N = 105 particles, whichis more than needed to produce very accurate results. Test are performed to ensurethat the time step ∆t is sufficiently small, and the simulation duration sufficientlylong, so that the associated numerical errors are negligible.

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Appendix B. Cloud-in-Cell Particle/Mesh Method

The cloud-in-cell (CIC) particle/mesh method is applicable to both the random-walk and the mean-drift models, and is similar to the method generally used inLES/PDF simulations.As in the mesh-free method, there are N computational particles. There is also a

mesh of uniform spacing h. There is a linear-spline basis function associated witheach mesh node. Means are estimated at the nodes using kernel estimation withthe basis functions as the kernels, and then the means are represented by linearsplines.The differences compared to the mesh-free method are:

(1) All means are estimated and represented using the CIC linear basis func-tions.

(2) The IEM mixing model is implemented directly using the CIC means Ω

and φ by integrating the equation

dφ∗

dt= −Ω∗(φ∗ − φ∗). (B1)

(3) The mean drift is implemented by

φ∗(t+∆t) = φ∗(t) + [φ(t+∆t)− φ(t)]∗, (B2)

where φ(x, t+∆t) is the solution after time ∆t of the partial differential equation

∂〈ρ〉φ

∂t=

∂

∂x

(〈ρ〉D

∂φ

∂x

), (B3)

from the initial condition φ(x, t) = φ(x, t). Equation B3 is integrated numericallyat the nodes, using a fully implicit three-point difference scheme.

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